This thesis considers the development of a cryogenic Transition Edge Sensor (TES) microcalorimeter for X-ray astronomy. An optimum configuration for this application has yet to be established and some of the design criteria and fabrication methods are discussed on Chapter 5. Results from investigations into different TES material combinations are presented in Chapter 6. Degradation due to interdiffusion between materials has an impact on the reliability of TES detectors and analysis of Al/Ag, Ti/Au, Cu/Nb and Ir/Nb systems has formed a large part of our recent work. Such comparisons show the relative stability of iridium-based detectors. The superconducting properties of thin film iridium samples were investigated in the context of detector development. The measured superconducting-to-normal transitions are presented in Chapter 7. Changes in transition temperature from 157 mK to 52 mK were observed. A model based on the formation of thin iridium silicide layers during film deposition is proposed to describe this effect. Chapter 8 contains results from testing of a Ti/Au single pixel TES device. The capability of our digital signal processing system was demonstrated with the acquisition of X-ray spectra with photopeak energy resolutions of 25 eV FWHM at 6 keV, close to the detector limit. Imaging spectrometers are required for astronomy applications and our extended bismuth absorber with distributed read-out concept is described in Chapter 9. Photon propagation through bismuth at low temperatures (~100 mK) has been modelled using a Monte Carlo approach, and the results of the simulations indicate that large square bismuth absorbers (of side length ~5 mm) with corner TES read-out can achieve spatial resolutions better than 250 m. This is adequate to satisfy the current specification for the narrow-field imaging spectrometer instrument of ESA's proposed XEUS mission.